The "Arab Spring" may not have succeeded in bringing democracy to the Middle East. But it has provided powerful evidence of a different phenomenon: the illusion of U.S. influence over governments we once considered our clients.

Take Egypt. Before 2011, the Bush and Obama administrations tried to nudge the autocratic Hosni Mubarak toward democracy; Mubarak ignored the advice. Last year, the Obama administration pleaded (gently) with the freely elected Mohamed Morsi to make his Muslim Brotherhood government more inclusive; Morsi ignored the advice. Now Egypt's armed forces have seized power and the United States is begging Gen. Abdel Fattah Sisi to refrain from cracking down too hard. Will he comply? Not likely. Whatever happened to our leverage as a superpower?

If the United States could be expected to have influence over any institution in the Arab world, it would be the government of Egypt, which collects $1.6 billion a year in American aid.

But two factors have diminished the leverage that the United States once gained by doling out foreign aid: less money and more competition.

First, $1.6 billion doesn't buy what it used to. U.S. aid to Egypt has been shrinking for most of Sisi's career. Adjusted for inflation, this year's $1.6 billion is only about one-third as much as the United States spent on Egypt aid in 1986.

The military portion of that annual aid, $1.3 billion, goes mostly to buying aircraft and tanks made in U.S. factories; the non-military portion, $250 million, is little more than a drop in the bucket for Egypt's sprawling economy. Sisi and other Egyptians know this all too well — but American politicians often sound as if they haven't noticed.

And other powers have stepped in to fill the breach. Last month, Saudi Arabia, the United Arab Emirates and Kuwait announced that they were rushing $12 billion in economic aid to Egypt to help the military regime stabilize the economy. Those gulf monarchies had an agenda too; they fear the Muslim Brotherhood, don't yearn for the restoration of democracy, and would be perfectly happy if Sisi cracked down hard. If foreign aid creates leverage, the sheikdoms' $12 billion trumps our $1.6 billion handily.

Besides, foreign influence in countries struggling toward democracy is a double-edged sword. The Obama administration has managed to alienate both sides in Egypt's political battle; the Muslim Brotherhood thinks the United States plotted to undermine it, and the military and its secular supporters say the United States is being too hard on Sisi now. All sides, including Sisi, cast themselves as nationalists; appearing to bow to U.S. wishes won't help the general maintain his sky-high popularity.

In any case, the stakes for Egyptians are too high for U.S. advice to count for much. "We're a sideshow," notes Steven A. Cook, an Egypt scholar at the Council on Foreign Relations. "If you're an Egyptian leader, one of your best political strategies right now is to stick it to Washington."

And that's precisely what Sisi is doing. "You turned your backs on the Egyptians, and they can't forget that," he told the Washington Post last week. "Talking a lot about aid and U.S. assistance really hurts our pride and dignity… [but] if the Americans want to cut assistance, they can."

The general's prickliness reflected another paradox: As leverage, foreign aid can be hard to use. Threatening to cut military aid may be a useful way to get Egyptian leaders' attention; but actually reducing military cooperation could hurt U.S. interests as much as Egypt's — not only in keeping peace with Israel, but also in counter-terrorism efforts across the Arab world. That's why the Obama White House quickly walked away from the temptation to call Sisi's takeover a coup and freeze U.S. aid. Instead, it was left to Secretary of State John F. Kerry to categorize the military's action this way: "In effect, they were restoring democracy."

For Egyptians, the message was familiar, if not entirely clear: For two decades, American politicians have threatened to pull aid if things didn't change, but they never dared pull the trigger. This time is unlikely to be much different.

The same limits apply, in different proportions, to U.S. policy in other parts of the Arab world. In 2012, Obama proposed major new funding to support nascent Arab democracies; most of the money never came through. "We're responding to events of historic consequence with the equivalent of the change we can find in the couch cushions," said Tamara Cofman Wittes of the Brookings Institution's Saban Center, who ran democracy aid programs at the State Department.

That doesn't mean the United States has no influence over events in the Arab world. It just means we have less sway than we often imagine — and the source of our influence may not be the size of the checks we write.

"We do have leverage, but it's not where we think it is," Wittes said. "What they want is international recognition. They want to be connected to the rest of the world. And they want our blessing. That may be our most effective leverage."

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